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Intuitionistic Fuzzy Set (IFS) can be used as a general tool for modeling problems of decision making under uncertainty where, the degree of rejection is defined simultaneously with the degree of acceptance of a piece of information in such a way that these degrees are not complement to each other. Accordingly, an attempt is made to solve intuitionistic fuzzy linear programming problems using a technique based on an earlier technique proposed by Zimmermann to solve fuzzy linear programming problem. Our proposed technique does not require the existing ranking of intuitionistic fuzzy numbers. This method is also different from the existing weight assignment method or the Angelov’s method. A comparative study is undertaken and interesting results have been presented.

Optimization problems exhibit some level of imprecisions and vagueness. Such phenomena have been well-captured through fuzzy sets in modeling these problems. Applications of fuzzy set theory in optimization of decisions have been studied extensively ever since the introduction of fuzzy sets. The theory of fuzzy sets proposed by Zadeh is a realistic and practical means to describe the objective world that we live in and has also been successfully applied in various other fields.

In decision making problems, the concept of maximizing decision under uncertainty was proposed by Bellman and Zadeh [

Recent years have witnessed a growing interest in the study of decision making problems under uncertainty with intuitionistic fuzzy sets/numbers [

Angelov [

In this paper, our aim is to propose a method to solve intuitionistic fuzzy linear programming problem (IFLPP) using a technique based on an earlier technique proposed by Zimmermann [

The paper is organized in six sections. The introductory section is followed by presentation of some basic concepts necessary for the development of a mechanism for solving intuitionistic fuzzy linear programming problems. In this section, basic concept of Triangular Intuitionistic Fuzzy Number (TIFN) is described. In Section 3, we discuss fuzzy linear programming problem and introduce a new method analogous with it, to solve IFLPP when both the coefficient matrix of the constraints and cost coefficients are intuitionistic fuzzy in nature. In Section 4, there is a comparative study between some of the other optimization techniques with our proposed technique for solving an intuitionistic fuzzy linear programming problem. Section 5 concludes the present paper and refers to some problems for further studies which is followed by a list of references in the last section.

Definition 1 [

A ˜ = { 〈 x j , μ A ˜ ( x j ) , ν A ˜ ( x j ) 〉 : x j ∈ U }

where the functions μ A ˜ : U → [ 0,1 ] and ν A ˜ : U → [ 0,1 ] respectively define the degree of membership and the degree of non-membership of an element x j ∈ U , such that they satisfy the following conditions:

0 ≤ μ A ˜ ( x j ) + ν A ˜ ( x j ) ≤ 1, ∀ x j ∈ U ;

known as intuitionistic condition. The degree of acceptance μ A ˜ ( x ) and of non-acceptance ν A ˜ ( x ) can be arbitrary.

Definition 2 [

Definition 3 [

Definition 4 [

Definition 5 [

Definition 6 [

A ˜ α , β = { x j ∈ U : μ A ˜ ( x j ) ≥ α , ν A ˜ ( x j ) ≤ β } .

Thus, the ( α , β ) -cut of an intuitionistic fuzzy set to be denoted by A ˜ ( α , β ) , is defined as the crisp set of elements x which belong to A ˜ at least to the degree α and which does not belong to A ˜ at most to the degree β .

Definition 7 [

1) an intuitionistic fuzzy subset of the real line ℜ ;

2) normal, i.e., ∃ x 0 ∈ ℜ such that μ A ˜ j ( x 0 ) = 1 , (so ν A ˜ j ( x 0 ) = 0 );

3) convex for the membership function, i.e.,

μ A ˜ j ( λ x 1 + ( 1 − λ ) x 2 ) ≥ min { μ A ˜ j ( x 1 ) , μ A ˜ j ( x 2 ) } ; ∀ x 1 , x 2 ∈ ℜ , λ ∈ [ 0 , 1 ] ;

4) concave for the non-membership function, i.e.,

ν A ˜ j ( λ x 1 + ( 1 − λ ) x 2 ) ≤ max { ν A ˜ j ( x 1 ) , ν A ˜ j ( x 2 ) } ; ∀ x 1 , x 2 ∈ ℜ , λ ∈ [ 0 , 1 ] .

Definition 8 [

μ A ˜ t ( x ) = { x − a + l l w a ; a − l ≤ x < a a + r − x r w a ; a ≤ x ≤ a + r 0 ; otherwise (1)

and

ν A ˜ t ( x ) = { ( a − x ) + u a ( x − a + l ) l ; a − l ≤ x < a ( x − a ) + u a ( a + r − x ) r ; a ≤ x ≤ a + r 1 ; otherwise (2)

where l, r are called spreads and a is called mean value. w a and u a represent the maximum degree of membership and minimum degree of non-membership respectively such that they satisfy the condition

0 ≤ w a ≤ 1, 0 ≤ u a ≤ 1 and 0 ≤ w a + u a ≤ 1.

Note 1.: From the above definitions we see that the numbers μ A ˜ ( x ) and ν A ˜ ( x ) reflect respectively the extent of a degree of acceptance and that of rejection of an element x to the set A ˜ , and the number π A ˜ ( x ) is the extent of indeterminacy.

We consider the linear programming problem (LPP) with cost of decision variables and co-efficient matrix of constraints represented as trapezoidal fuzzy in nature:

〈 c ˜ , x 〉 = f j ( x k ) = f j ( x ) = max Z ˜ = ∑ k = 1 n c ˜ k x k

subject to

∑ k = 1 n A ˜ j k x k ≤ B ˜ j , 1 ≤ j ≤ m ; x k ≥ 0

where,

μ A ˜ j k ( x ) = { 1 ; for x < a j k a j k + d j k − x d j k ; for a j k ≤ x ≤ a j k + d j k 0 ; for x ≥ a j k + d j k (3)

μ B ˜ j ( x ) = { 1 ; for x < b j b j + p j − x p j ; for b j ≤ x ≤ b j + p j 0 ; for x ≥ b j + p j (4)

μ c ˜ k ( x ) = { 1 ; for x < α k α k + β k − x β k ; for α k ≤ x ≤ α k + β k 0 ; for x ≥ α k + β k (5)

The class of fuzzy linear programming models is not uniquely defined as it depends upon the type of fuzziness as also its specification as prescribed by the decision maker. Accordingly, the class of FLPP can be broadly classified as:

1) LPP with fuzzy inequalities and crisp objective function,

2) LPP with crisp inequalities and fuzzy objective function,

3) LPP with fuzzy inequalities and fuzzy objective function,

4) LPP with fuzzy resources and fuzzy coefficients, also termed as LPP with fuzzy parameters, i.e., elements of c , A , B are fuzzy numbers.

Intuitionistic fuzzy optimization (IFO), a method of uncertainty optimization, is put forward on the basis of intuitionistic fuzzy sets, due to Atanassov [

In this paper, we try to examine the case in which all the co-efficients and the right hand side constants appearing in the constraints are modeled as TIFN and then reformulated as a LPP with intuitionistic fuzzy inequalities and objective function.

max Z ˜ = ∑ k = 1 n c ˜ k I x k

subject to

∑ k = 1 n A ˜ j k I x k ≤ ˜ B ˜ j I ; 1 ≤ j ≤ m ; x k ≥ 0 for 1 ≤ k ≤ n

where,

c ˜ k I = 〈 c k , l k , r k ; ν k , μ k 〉 , A ˜ j k I = 〈 a j k , l j k , r j k ; ν j k , μ j k 〉 , B ˜ j I = 〈 b j , l j , r j ; ν j , μ j 〉 and x = ( x 1 , x 2 , ⋯ , x n ) ′ .

As co-efficients are TIFN so maximum membership occurs at c k , a j k and b j , hence the above given problem is reformulated as

J = max Z ˜ = ∑ k = 1 n c k x k

subject to

∑ k = 1 n a j k x k ≤ ˜ b j , 1 ≤ j ≤ m ; x ≥ 0

where the inequality relations ≤ ˜ are considered as intuitionistic fuzzy inequalities.

For the objective function, the intuitionistic fuzzifier max is understood in the sense of the satisfaction of the aspiration level Z 0 as best as possible. To solve this we first choose an appropriate membership and non-membership function for each of the intuitionistic fuzzy inequality. In particular, μ 0 and ν 0 denote respectively the membership and non-membership functions for the objective function and μ j , ν j ( j = 1 , 2 , ⋯ , m ) respectively denote the membership and non-membership function for the j^{th} constraint. Let^{th} constraint. Then, we decide

This leads to the following equivalent crisp LPP:

where

Now, the above can be solved easily by using usual simplex method.

Thus, the following are the steps proposed to solve the LPP under the intuitionistic fuzzy environment

Algorithm:

Input: An Intuitionistic fuzzy LPP in mathematical form.

Output: Optimal solution and corresponding decision.

Step 1: Choose an aspiration level

Step 2: Defuzzify the intuitionistic fuzzy sets appearing as co-efficients in the objective and the constraints and subsequently rewrite the system with crisp numbers and intuitionistic inequalities.

Step 3: Construct membership function

Step 4: Taking minimal acceptance degree

where

Step 5: Accordingly, the above formulation is equivalent to:

Step 6: Solve the ordinary linear programming problem using simplex technique.

Example 1: Let us consider an intuitionistic fuzzy LPP as in the following:

System (14) is defuzzified into the crisp model as,

where

According to Zimmermann’s approach for a symmetric model, we assume that

are as given below:

Following Zimmermann’s approach to solve (14), we need to solve the following crisp LPP:

where

Accordingly, we have the formulation:

Now, by using simplex algorithm, we solve the above problem and obtain the solution as

For different values of

Sr. No. | |||||||||
---|---|---|---|---|---|---|---|---|---|

1 | 66 | 12 | 14 | 16 | 0.492663 | 0.376087 | 0.0 | 2.019565 | 60.58695 |

2 | 66 | 12 | 15 | 17 | 0.577319 | 0.357974 | 0.0 | 2.030928 | 60.92784 |

3 | 66 | 12 | 20 | 20 | 0.633928 | 0.311071 | 0.0 | 2.053571 | 61.60713 |

4 | 66 | 12 | 22 | 23 | 0.677165 | 0.275008 | 0.0 | 2.070866 | 62.12598 |

5 | 66 | 12 | 25 | 25 | 0.700729 | 0.255270 | 0.0 | 2.080292 | 62.40876 |

6 | 66 | 12 | 27 | 28 | 0.730263 | 0.230451 | 0.0 | 2.092105 | 62.76315 |

7 | 66 | 12 | 30 | 30 | 0.746913 | 0.216419 | 0.0 | 2.098765 | 62.96295 |

8 | 66 | 12 | 32 | 33 | 0.768361 | 0.198305 | 0.0 | 2.107345 | 63.22035 |

9 | 66 | 12 | 35 | 35 | 0.780748 | 0.187822 | 0.0 | 2.112299 | 63.36897 |

10 | 66 | 12 | 37 | 38 | 0.797029 | 0.174022 | 0.0 | 2.118812 | 63.56436 |

11 | 66 | 12 | 40 | 40 | 0.806603 | 0.165896 | 0.0 | 2.122642 | 63.67926 |

12 | 66 | 12 | 42 | 43 | 0.819383 | 0.155035 | 0.0 | 2.127753 | 63.83259 |

13 | 66 | 12 | 45 | 45 | 0.827004 | 0.148551 | 0.0 | 2.130802 | 63.92406 |

14 | 66 | 12 | 47 | 48 | 0.837301 | 0.139781 | 0.0 | 2.134921 | 64.04763 |

15 | 66 | 12 | 50 | 50 | 0.843511 | 0.134488 | 0.0 | 2.137405 | 64.12215 |

16 | 66 | 12 | 55 | 55 | 0.857142 | 0.122857 | 0.0 | 2.142857 | 64.28571 |

17 | 66 | 12 | 60 | 60 | 0.868589 | 0.113076 | 0.0 | 2.147436 | 64.42308 |

18 | 66 | 12 | 65 | 65 | 0.878383 | 0.104738 | 0.0 | 2.151335 | 64.54005 |

19 | 66 | 12 | 70 | 70 | 0.886740 | 0.097545 | 0.0 | 2.154696 | 64.64088 |

20 | 66 | 12 | 75 | 75 | 0.894056 | 0.091276 | 0.0 | 2.157623 | 64.72869 |

21 | 66 | 12 | 80 | 80 | 0.900485 | 0.085764 | 0.0 | 2.160194 | 64.80582 |

22 | 66 | 12 | 85 | 85 | 0.906178 | 0.080880 | 0.0 | 2.162471 | 64.87413 |

23 | 66 | 12 | 90 | 90 | 0.911255 | 0.076522 | 0.0 | 2.164502 | 64.93506 |

24 | 66 | 12 | 95 | 95 | 0.915811 | 0.072609 | 0.0 | 2.166324 | 64.98972 |

25 | 66 | 12 | 100 | 100 | 0.919921 | 0.069078 | 0.0 | 2.167969 | 65.03907 |

In Section 3, instead of IFLPP if we take the corresponding FLPP and solve the same using Zimmermann’s technique [

Problem 1:

Problem 2:

Problem 3:

Result and discussion:

In fact, there are some cases where due to insufficiency in the available information, the evaluation of the membership and non-membership functions together gives better and/or satisfactory result than considering either the membership value or the non-membership value. Accordingly, there remains a part indeterministic on which hesitation survives. Certainly fuzzy optimization is unable to deal such hesitation since in this case here membership and non-membership functions are complement to each other. Here, we extend Zimmermann’s optimization technique for solving FLPP. In our proposed technique, sum of membership degree and non-membership degree always taken as strictly less than one and hence hesitation arises. Consequently, to achieve the aspiration level

In human decision making problem, IFO plays an important and useful role. This approach converts the introduced intuitionistic fuzzy optimization (IFO) problem into a crisp (non-fuzzy) LPP. The advantage of the IFO problem is two-fold: they give a rich apparatus for formulation of optimization problems and, on the other hand, the solution of IFO problems can satisfy the objective(s) with a greater degree than the analogous fuzzy optimization problem.

Proposed technique | ||
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Zimmermann’s technique | ||

[ | ||

Weight assignment technique [ | ||

Angelov’s technique [ |

Proposed technique | ||
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Zimmermann’s technique | ||

Weight assignment technique | ||

Angelov’s technique |

Proposed technique | ||
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Zimmermann’s technique | ||

Weight assignment technique | ||

Angelov’s technique |

There is considerable scope for research in this domain. In future, this research work could be extended to some other uncertain environment such as representation using Pythagorean fuzzy set [

This research is supported by UGC SAP-DRS Phase-III programme at the Department of Mathematics. UGC’s financial support is highly appreciated.

The authors declare no conflicts of interest regarding the publication of this paper.

Kabiraj, A., Nayak, P.K. and Raha, S. (2019) Solving Intuitionistic Fuzzy Linear Programming Problem. International Journal of Intelligence Science, 9, 44-58. https://doi.org/10.4236/ijis.2019.91003